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Properties

Label 2-945-945.769-c0-0-1
Degree $2$
Conductor $945$
Sign $0.230 + 0.973i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.499 + 0.866i)9-s + (0.266 − 0.223i)11-s + (0.766 + 0.642i)12-s + (0.266 − 1.50i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 − 1.62i)17-s + (−0.939 − 0.342i)20-s + (0.173 + 0.984i)21-s + (0.173 + 0.984i)25-s + 0.999·27-s + 28-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)7-s + (−0.499 + 0.866i)9-s + (0.266 − 0.223i)11-s + (0.766 + 0.642i)12-s + (0.266 − 1.50i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 − 1.62i)17-s + (−0.939 − 0.342i)20-s + (0.173 + 0.984i)21-s + (0.173 + 0.984i)25-s + 0.999·27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6711877571\)
\(L(\frac12)\) \(\approx\) \(0.6711877571\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
good2 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.932193259647158097920490805422, −9.480937518305125918181735572853, −8.222555237789267422187073501005, −7.50897873953630043937730376114, −6.65643549473945309232452587171, −5.74513806165760813501631504565, −5.12975998245539599460342218697, −3.47500043269904499151260195264, −2.74422810130653709946060285108, −0.77332851815689794909817116420, 1.53801134842632792938352404797, 3.50985783305189923381743723698, 4.29270508110095679243409894390, 5.20489217602122366970862629276, 5.99602603643895674184690048409, 6.56494481577358807462712622087, 8.420646247563117095924346236233, 9.023141999426701466367214606966, 9.649582547625254727554018977389, 10.07674315290485986100951367994

Graph of the $Z$-function along the critical line